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Derivative of:
Derivative of y(x)=sqrt(3x+5)
Derivative of y=tan^24x
Derivative of y=sinx/x^4
Derivative of y=sin^2x+3x^-4
Identical expressions
y=sin^2x+3x^- four
y equally sinus of squared x plus 3x to the power of minus 4
y equally sinus of squared x plus 3x to the power of minus four
y=sin2x+3x-4
y=sin²x+3x^-4
y=sin to the power of 2x+3x to the power of -4
Similar expressions
y=sin^2x-3x^-4
y=sin^2x+3x^+4

The derivative of the function/ y=sin^2x+3x^-4

Derivative of y=sin^2x+3x^-4

The solution

You have entered [src]

   2      3 
sin (x) + --
           4
          x

$$\sin^{2}{\left(x \right)} + \frac{3}{x^{4}}$$

sin(x)^2 + 3/x^4

Detail solution

Differentiate $\sin^{2}{\left(x \right)} + \frac{3}{x^{4}}$ term by term:
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $2 \sin{\left(x \right)} \cos{\left(x \right)}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $\frac{1}{x^{4}}$ goes to $- \frac{4}{x^{5}}$
  So, the result is: $- \frac{12}{x^{5}}$
The result is: $2 \sin{\left(x \right)} \cos{\left(x \right)} - \frac{12}{x^{5}}$
Now simplify:

$\sin{\left(2 x \right)} - \frac{12}{x^{5}}$

The answer is:

$\sin{\left(2 x \right)} - \frac{12}{x^{5}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  12                  
- -- + 2*cos(x)*sin(x)
   5                  
  x

$$2 \sin{\left(x \right)} \cos{\left(x \right)} - \frac{12}{x^{5}}$$

Simplify

The second derivative [src]

  /   2         2      30\
2*|cos (x) - sin (x) + --|
  |                     6|
  \                    x /

$$2 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} + \frac{30}{x^{6}}\right)$$

Simplify

The third derivative [src]

   /45                \
-8*|-- + cos(x)*sin(x)|
   | 7                |
   \x                 /

$$- 8 \left(\sin{\left(x \right)} \cos{\left(x \right)} + \frac{45}{x^{7}}\right)$$

Simplify

The graph

Derivative of y=sin^2x+3x^-4